On the second ascent of Sinai, God proclaimed His own Name to Moses in a formula that Jews have recited in every generation since.
Targum Pseudo-Jonathan, the Aramaic paraphrase of the Torah, preserves the opening of the Thirteen Attributes. "The Lord made His Shekhinah to pass by before his face, and proclaimed, The Lord, the Lord God, merciful and gracious, long-suffering, and nigh in mercies, abounding to exercise compassion and truth" (Exodus 34:6).
This is the passage that Jews chant on Yom Kippur, on fast days, during Selichot in the weeks before the High Holidays. It is, according to the sages, the core formula for reaching divine forgiveness. "Whenever Israel sins, let them perform this order before Me, and I will forgive them," the Talmud teaches God saying (Rosh Hashanah 17b). The order is these words, in this sequence.
Notice what the Targum emphasizes. Rachman v'chanun, merciful and gracious. Rachiq regaz, slow to anger, literally "far from wrath." Mekarav rachamin, near in mercies. The portrait is of a God whose first instinct is compassion and whose wrath is the exception, distant and delayed.
The camp below had danced around a calf. The Lord above responded not with thunder but with this liturgy. The calf did not cause God to become angry. It caused Him to reveal mercy.
Takeaway: The deepest Name of God is not judgment but compassion. That Name is available to every Jew who chants the Thirteen Attributes with honest intention.